bioRxiv preprint doi: https://doi.org/10.1101/2020.12.10.419200; this version posted December 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 1 Resonating neurons stabilize heterogeneous grid-cell networks 2 3 Divyansh Mittal and Rishikesh Narayanan* 4 5 Cellular Neurophysiology Laboratory, Molecular Biophysics Unit, Indian Institute of 6 Science, Bangalore, India. 7 8 *Corresponding Author 9 10 Rishikesh Narayanan, Ph.D. 11 Molecular Biophysics Unit 12 Indian Institute of Science 13 Bangalore 560 012, India. 14 15 e-mail: [email protected] 16 Phone: +91-80-22933372 17 Fax: +91-80-23600535 18 Number of words: 250 (abstract), 120 (significance statement) 19 20 Abbreviated title: Resonating neurons stabilize heterogeneous networks 21 22 Keywords: grid cells, entorhinal cortex, continuous attractor network, resonance, 23 heterogeneities 24 25 Author contributions 26 D. M. and R. N. designed experiments; D. M. performed experiments and carried out data 27 analysis; D. M. and R. N. co-wrote the paper. 28 29 Competing financial interests 30 The authors declare no conflict of interest. 31 32 Acknowledgments 33 This work was supported by the Wellcome Trust-DBT India Alliance (Senior fellowship to 34 RN; IA/S/16/2/502727), Human Frontier Science Program (HFSP) Organization (RN), the 35 Department of Biotechnology through the DBT-IISc partnership program (RN), the Revati 36 and Satya Nadham Atluri Chair Professorship (RN), and the Ministry of Human Resource 37 Development (RN & DM). The authors thank Dr. Poonam Mishra, Dr. Sufyan Ashhad and 38 the members of the cellular neurophysiology laboratory for helpful discussions and for 39 comments on a draft of this manuscript. The authors thank Dr. Ila Fiete for helpful 40 discussions. 41 42 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.10.419200; this version posted December 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 1 ABSTRACT 2 3 Grid cells in the medial entorhinal cortex manifest multiple firing fields, patterned to 4 tessellate external space with triangles. Although two-dimensional continuous attractor 5 network (CAN) models have offered remarkable insights about grid-patterned activity 6 generation, their functional stability in the presence of biological heterogeneities remains 7 unexplored. In this study, we systematically incorporated three distinct forms of intrinsic and 8 synaptic heterogeneities into a rate-based CAN model driven by virtual trajectories, 9 developed here to mimic animal traversals and improve computational efficiency. We found 10 that increasing degrees of biological heterogeneities progressively disrupted the emergence of 11 grid-patterned activity and resulted in progressively large perturbations in neural activity. 12 Quantitatively, grid score and spatial information associated with neural activity reduced 13 progressively with increasing degree of heterogeneities, and perturbations were primarily 14 confined to low-frequency neural activity. We postulated that suppressing low-frequency 15 perturbations could ameliorate the disruptive impact of heterogeneities on grid-patterned 16 activity. To test this, we formulated a strategy to introduce intrinsic neuronal resonance, a 17 physiological mechanism to suppress low-frequency activity, in our rate-based neuronal 18 model by incorporating filters that mimicked resonating conductances. We confirmed the 19 emergence of grid-patterned activity in homogeneous CAN models built with resonating 20 neurons and assessed the impact of heterogeneities on these models. Strikingly, CAN models 21 with resonating neurons were resilient to the incorporation of heterogeneities and exhibited 22 stable grid-patterned firing, through suppression of low-frequency components in neural 23 activity. Our analyses suggest a universal role for intrinsic neuronal resonance, an established 24 mechanism in biological neurons to suppress low-frequency neural activity, in stabilizing 25 heterogeneous network physiology. 26 2 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.10.419200; this version posted December 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 1 SIGNIFICANCE STATEMENT 2 3 A central theme that governs the functional design of biological networks is their ability to 4 sustain stable function despite widespread parametric variability. However, several 5 theoretical and modeling frameworks employ unnatural homogeneous networks in assessing 6 network function owing to the enormous analytical or computational costs involved in 7 assessing heterogeneous networks. Here, we investigate the impact of biological 8 heterogeneities on a powerful two-dimensional continuous attractor network implicated in the 9 emergence of patterned neural activity. We show that network function is disrupted by 10 biological heterogeneities, but is stabilized by intrinsic neuronal resonance, a physiological 11 mechanism that suppresses low-frequency perturbations. As low-frequency perturbations are 12 pervasive across biological systems, mechanisms that suppress low-frequency components 13 could form a generalized route to stabilize heterogeneous biological networks. 14 3 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.10.419200; this version posted December 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 1 INTRODUCTION 2 Stability of network function, defined as the network’s ability to elicit robust functional 3 outcomes despite perturbations to or widespread variability in its constitutive components, is 4 a central theme that governs the functional design of several biological networks. Biological 5 systems exhibit ubiquitous parametric variability spanning different scales of organization, 6 quantified through statistical heterogeneities in the underlying parameters. Strikingly, in spite 7 of such large-scale heterogeneities, outputs of biological networks are stable, and are 8 precisely tuned to meet physiological demands. A central question that spans different scales 9 of organization is on the ability of biological networks to achieve physiological stability in 10 the face of ubiquitous parametric variability (1-11). 11 Biological heterogeneities are known to play critical roles in governing stability of 12 network function, through intricate and complex interactions among mechanisms underlying 13 functional emergence (8, 9, 11-29). However, an overwhelming majority of theoretical and 14 modeling frameworks lack the foundation to evaluate the impact of such heterogeneities on 15 network output, as they employ unnatural homogeneous networks in assessing network 16 function. The paucity of heterogeneous network frameworks is partly attributable to the 17 enormous analytical or computational costs involved in assessing heterogeneous networks. In 18 this study, we quantitatively address questions on the impact of distinct forms of biological 19 heterogeneities on the functional stability of a two-dimensional continuous attractor network 20 (CAN), which has been implicated in the generation of patterned neuronal activity in grid 21 cells of the medial entorhinal cortex (30-36). Although the continuous attractor framework 22 has offered insights about information encoding across several neural circuits (27, 30, 34, 36- 23 48), the fundamental question on the stability of 2-D CAN models in the presence of 24 biological heterogeneities remains unexplored. Here, we systematically assessed the impact 25 of biological heterogeneities on stability of emergent spatial representations in a 2-D CAN 26 model, and unveiled a physiologically plausible neural mechanism that promotes stability 27 despite the expression of heterogeneities. 28 We first developed an algorithm to generate virtual trajectories that closely mimicked 29 animal traversals in an open arena, to provide better computational efficiency in terms of 30 covering the entire arena within shorter time duration. We employed these virtual trajectories 31 to drive a rate-based homogeneous CAN model that elicited grid-patterned neural activity 32 (30), and systematically introduced different degrees of three distinct forms of biological 33 heterogeneities. The three distinct forms of biological heterogeneities that we introduced, 34 either individually or together, were in neuronal intrinsic properties, in afferent inputs 4 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.10.419200; this version posted December 11, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 1 carrying behavioral information and in local-circuit synaptic connectivity. We found that the 2 incorporation of these different forms of biological heterogeneities disrupted the emergence 3 of grid-patterned activity by introducing perturbations in neural activity, predominantly in 4 low-frequency components. In the default model where neurons were integrators, grid- 5 patterns and spatial information in neural activity were progressively lost with increasing 6 degrees of biological heterogeneities, and were accompanied